Lusternik–Schnirelmann category of products with half-smashes

We show that for a fixed space X and any sufficiently highly connected space A (conn(A) > dim(X) is more than enough), the Lusternik-Schnirelmann category of products with X is remarkably stable with respect to changes in the second variable: cat(X x A) = cat(X x (A (sic) B)) for all spaces B. Taking X = S-n leads to a closure property for the collections of spaces which do or do not satisfy the Ganea condition cat(S-n x A) = 1 + cat(A).